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A Friedman number is an integer, which represented in a given numeral system, is the result of a non trivial expression using all its own digits in combination with any of the four basic arithmetic operators (+, −, ×, ÷), additive inverses, parentheses, exponentiation, and concatenation. Here, non trivial means that at least one operation besides concatenation is used. Leading zeros cannot be used, since that would also result in trivial Friedman numbers, such as 024 = 20 + 4. For example, 347 is a Friedman number in the decimal numeral system, since 347 = 73 + 4. The decimal Friedman numbers are:

25, 121, 125, 126, 127, 128, 153, 216, 289, 343, 347, 625, 688, 736, 1022, 1024, 1206, 1255, 1260, 1285, 1296, 1395, 1435, 1503, 1530, 1792, 1827, 2048, 2187, 2349, 2500, 2501, 2502, 2503, 2504, 2505, 2506, 2507, 2508, 2509, 2592, 2737, 2916,
Friedman numbers are named after Erich Friedman, a now retired mathematics professor at Stetson University and recreational mathematics enthusiast. A Friedman prime is a Friedman number that is also prime. The decimal Friedman primes are:

127, 347, 2503, 12101, 12107, 12109, 15629, 15641, 15661, 15667, 15679, 16381, 16447, 16759, 16879, 19739, 21943, 27653, 28547, 28559, 29527, 29531, 32771, 32783, 35933, 36457, 39313, 39343, 43691, 45361, 46619, 46633, 46643, 46649, 46663, 46691, 48751, 48757, 49277, 58921, 59051, 59053, 59263, 59273, 64513, 74353, 74897, 78163, 83357, .

There usually are fewer 2 digit Friedman numbers than 3 digit and more in any given base, but the 2 digit ones are easier to find. If we represent a 2 digit number as mb + n, where b is the base and m, n are integers from 0 to b−1, we need only check each possible combination of m and n against the equalities mb + n = mn, and mb + n = nm to see which ones are true. We need not concern ourselves with m + n or m × n, since these will always be smaller than mb + n when n < b. The same clearly holds for m − n and m / n.

Friedman numbers also exist for bases other than base 10. For example, 110012 = 25 is a Friedman number in the binary numeral system, since 11001 = 10110. The first few known Friedman numbers in other small bases are shown below, written in their respective bases. Numbers shown in bold are nice Friedman numbers. base Friedman numbers211001, 11011, 111111, 1001111, 1010001, 3121, 221, 1022, 1122, 1211, 4121, 123, 1203, 1230, 1321, 5121, 224, 1232, 1241, 1242, 624, 52, 121, 124, 133, 7121, 143, 144, 264, 514, 833, 121, 125, 143, 251, 9121, 134, 314, 628, 1304, 11121, 2A9, 603, 1163, 1533, 12121, 127, 135, 144, 163, 13121, 237, 24A, 1245, 1246, 14121, 128, 135, 144, 173, 1526, 121, 136, 154, 336, 16121, 129, 145, 183, 27D,

In a trivial sense, all Roman numerals with more than one symbol are Friedman numbers. The expression is created by simply inserting + signs into the numeral, and occasionally the − sign with slight rearrangement of the order of the symbols. Some research into Roman numeral Friedman numbers for which the expression uses some of the other operators has been done. The first such nice Roman numeral Friedman number discovered was 8, since VIII = (V I) × II. Other such nontrivial examples have been found. The difficulty of finding nontrivial Friedman numbers in Roman numerals increases not with the size of the number (as is the case with positional notation numbering systems) but with the numbers of symbols it has. For example, it is much tougher to figure out whether 147 (CXLVII) is a Friedman number in Roman numerals than it is to make the same determination for 1001 (MI). With Roman numerals, one can at least derive quite a few Friedman expressions from any new expression one discovers. Since 8 is a nice nontrivial nice Roman numeral Friedman number, it follows that any number ending in VIII is also such a Friedman number.